Topological quantum error correction codes are known to be able to tolerate arbitrary local errors given sufficient qubits. This includes correlated errors involving many local qubits. In this work, we quantify this level of tolerance, numerically studying the effects of many-qubit errors on the performance of the surface code. We find that if increasingly large-area errors are at least moderately exponentially suppressed, arbitrarily reliable quantum computation can still be achieved with practical overhead. We furthermore quantify the effect of nonlocal two-qubit correlated errors, which would be expected in arrays of qubits coupled by a polynomially decaying interaction, and when using many-qubit coupling devices. We surprisingly find that the surface code is very robust to this class of errors, despite a provable lack of a threshold error rate when such errors are present.
We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum error-correction period by studying when the single-qubit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a two-dimensional Ohmic bath, the time available per quantum error-correction period vanishes in the thermodynamic limit of a large code size L due to induced superluminal interactions, although it does so only like 1/lnL. For all other bath types considered, this time remains finite as $L\ensuremath{\rightarrow}\ensuremath{\infty}$.
Surface codes are promising for practical quantum error correction due to their high threshold and experimental feasibility. However, their performance under realistic noise conditions, particularly those involving correlated errors, requires further investigation. In this study, we systematically investigate the effect of several distinct types of correlated errors that could potentially arise from next-nearest-neighbor coupling in quantum systems. Our analytical results reveal that these correlated errors exhibit lower logical error rates due to their inherent structural symmetries. We further perform circuit-level simulations that incorporate realistic circuit level errors alongside correlated noise components, demonstrating that surface code performance is intimately related to the structural characteristics of the correlation patterns. This deepens our insights into the threshold of surface code and hence facilitates a more robust design of quantum circuits with a higher noise threshold.